Families of curves in Vinberg representations

Inspired by orbit parametrizations in arithmetic statistics, we explain how to construct families of curves associated to certain nilpotent elements in $\mathbb{Z}/m\mathbb{Z}$-graded Lie algebras, generalizing work of Thorne to the $m\geq 3$ case and the non-simply laced case. We classify such families arising from subregular nilpotents in stable gradings and interpret almost all orbit parametrizations associated with algebraic curves appearing in the literature in this framework. As an extended example, we give a Lie-theoretic proof of the integral orbit parametrization of $5$-Selmer elements of elliptic curves over $\mathbb{Q}$, using a $\mathbb{Z}/5\mathbb{Z}$-grading on a Lie algebra of type $E_8$.